In this poem, Moore writes about the thoughts and memories that crowd his mind before he falls asleep at night. The phrase in question means that these memories come to him before--"ere"--he falls asleep--in other words, before "slumber" overtakes him. When he says sleep or slumber's chain has bound him, he refers to the way sleep holds us in place and controls our thoughts. What Moore focuses on is the liminal space between being absorbed by all the worries and tasks of daily life and being "dead to the world" in sleep.
In those moments before sleep, the past comes flooding into Moore's mind. At first, good--"fond"-- memories of his boyhood come back to him. But then, as he remembers how heartbreak and death have hit people he knew in happier days, the memories become darker.
In the second stanza, he likens the friends he once knew, all linked together as if to one tree, to fallen leaves. The old days are gone and won't come back, so "sad memory" fills the poet's mind until he drifts off into slumber's chains.